Abstract

Based on the characterizations of reproducing cones, some fixed point theorems for multi-valued increasing, decreasing, and mixed monotone operators are established. As an application, the solvability of a fractional integral inclusion is discussed.

Keywords

1 Introduction

Single-valued monotone operators have been widely investigated. The results on the existence of fixed points for single-valued monotone operators are bounteous and successful and have found various applications to nonlinear integral equations and differential equations. For details, we can refer to [1–4] and the references therein.

It is well known that mixed monotone operators were introduced by Guo and Lakshmikantham [5] in 1987. Later, Bhaskar and Lakshmikantham [6] introduced the notation of coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial order. Their study not only has important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology, economics, etc. (see [5–13] and the references therein).

Very recently, Harjani, Lopez and Sadarangani [14] have established the existence results of coupled fixed point for mixed monotone operators and further obtained their applications to integral equations. Then in [15], Bu, Feng, and Li extended the study to mixed monotone ternary operators.

It is natural to extend these studies to the multi-valued case. In 1984, Nishniannidze [16] introduced the multi-valued monotone operators. Nguyen and Nguyen [17] investigated the fixed points for multi-valued increasing operators and then in [18], a fixed point theorem for a multi-valued increasing operator was established and applied to a discontinuous elliptic equation. Huang and Fang [19] extended the mixed monotone operators to the multi-valued case, the obtained result was applied to a class of integral inclusions. For some recent fixed point theorems for a multi-valued monotone operator, a mixed monotone operator, and their applications in differential equations and differential inclusions, we can refer to [19–25] and the references therein for details. All these papers need the existence of a lower or an upper solution to the operator inclusion.

Motivated by the above work, in this paper, we shall use the properties of reproducing cones to establish some fixed point theorems for multi-valued monotone and mixed monotone operators. Compared with the previous work, we remove the requirement of the existence of a lower or upper solution. When the operator is single-valued and mixed monotone, we get an existence and uniqueness result.

This paper is organized as follows: in Section 2, some basic knowledge and the properties of a reproducing cone are presented; then in Section 3, the existence of a fixed point for multi-valued monotone operators is established; in Section 4, coupled fixed point theorems are presented for single-valued and multi-valued mixed monotone operators. In the final section, as applications of our results, the solvability of a fractional integral inclusion is discussed.

2 Preliminaries

In this section, we recall some standard definitions and notations needed in the following section. For convenience of the reader, we suggest that one refers to [3, 26] for details.

At the beginning of this section, let us recall some concepts of the theory of cones in Banach spaces. These concepts play an important role in the remainder of this paper.

Let X be a Banach space, a closed convex set \(P\subset X\) is called a cone, if \(x\in P\) and \(x\neq0\) implies \(\alpha x\in P\) for \(\alpha\geq0\) and \(\alpha x\notin P\) for \(\alpha< 0\). A cone defines a partial order in the Banach space X: we write \(x\leq y\) or \(y\geq x\) if \(y-x\in P\). The relation enjoys the following properties: inequalities may be multiplied by a nonnegative numbers; inequalities of the same kind may be added by terms; one may pass to the limit in inequalities; \(x\leq y\) and \(y\leq x\) implies \(x=y\). Denote \(-P=\{-x\mid x\in P\}\), then −P is a cone too.

It is well known that if X be a partially ordered Banach space endowed with partial order ≤, then the subset \(P=\{x\in X\mid0\leq x\}\) is a cone.

Definition 2.1

A cone P is called normal, if there is a \(K>0\), such that \(0\leq x\leq y\) implies \(\Vert x\Vert\leq K\Vert y\Vert\) and K does not depend on x and y. Any such \(K=K(P)\) is called a normal constant of P. A cone P is called reproducing, if each \(x\in X\) admits a presentation \(x=u-v\) (\(u,v\in P\)). The elements u, v are, of course, not unique.

Definition 2.5

(1)

Let \(T:X\rightarrow2^{X}\setminus\{\phi\}\) be a multi-valued operator, \(\overline{x}\in X\) is called a fixed point of T, if \(\overline{x}\in T(\overline{x})\).

(2)

Assume \(T : X\times X\rightarrow2^{X}\setminus\{\phi\}\) is a multi-valued operator, \((\overline{x},\overline{y})\in X\times X\) is called a coupled fixed point of T, if \(\overline{x}\in T(\overline {x},\overline{y})\) and \(\overline{y}\in T(\overline{y},\overline{x})\).

3 Fixed points for multi-valued monotone operators

In this section, some fixed point theorems for multi-valued increasing and decreasing operators are proved in partial ordered Banach space.

Throughout this paper, we assume that X is a Banach space, P is a normal and reproducing cone in X and the partial order ‘≤’ is induced by the cone P.

Theorem 3.1

Suppose the multi-valued operator\(T : X\rightarrow2^{X}\setminus\{\phi\}\)satisfies the following conditions:

Remark 3.2

1.

In Theorem 3.1, the assumptions (2)(i) and (2)(ii) imply that \(T(x)\prec T(y)\) for \(x\leq y\), i.e.T is a multi-valued increasing operator.

2.

It should be noticed that Theorem 3.1 cannot ensure the uniqueness of a fixed point. For example, let \(X=R\), \(\|x\|=|x|\), \(P=\{ x\in R\mid x\geq0\}\), then X is a Banach space and P a normal and reproducing cone.

All conditions of Theorem 3.1 are satisfied. It is easy to verify that the fixed point set of T is the interval \([-1,1]\).

3.

In the special case when T is single-valued, assumption (1) of Theorem 3.1 is naturally satisfied. Assumption (2) is simplified thus: there exists a linear operator \(L:X\rightarrow X\) with spectral radius \(r(L)<1\), \(L(P)\subset P\) such that

$$T(y)-T(x)\leq L(y-x),\quad \forall x\leq y. $$

In this case, for the fixed point of T we have existence and uniqueness; see [4].

Remark 3.3

In Theorem 3.1, assumptions (1), (2) are sufficient, without one of them, the existence of a fixed point cannot be ensured. For example, let \(X=R\), \(\|x\|=|x|\), \(P=\{x\in R\mid x\geq0\}\), then X is a Banach space and P a normal and reproducing cone.

For arbitrary \((x,y)\in X \times X\), since P is reproducing, there exist \(u_{1},v_{1},u_{2},v_{2}\in P\), such that \(x=u_{1}-v_{1}\), \(y=u_{2}-v_{2}\), or equivalently, \(y=(-v_{2})-(-v_{1})\). Then

Let \(X=C[0,1]\). For \(x\in X\), let \(\|x\|=\max_{t\in[0,1]}|x(t)|\), \(P=\{ x\in X\mid x(t)\geq0, t\in[0,1]\}\), then \((X,\Vert\cdot\Vert)\) is a Banach space and P is a normal and reproducing cone in X.

Let \(0<\alpha<1\) be a constant. For \(x\in X\), the Riemann-Liouville fractional integral operator \(I^{\alpha}\) is defined as follows [27]:

where \(L(x)(t)= M\int_{0}^{t}{(t-s)}^{\alpha-1}x(s)\,ds\). By Lemma 5.2, \(r(L)=0\) and \(L(P)\subset P\). Due to Theorem 3.1, T has a fixed point in x, i.e., the fractional differential inclusion has a solution.

This ends the proof. □

Declarations

Acknowledgements

We are grateful to the anonymous referees for their helpful suggestions. This research is supported in part by the Doctoral Fund of Education Ministry of China (20134219120003) and the Key Program of Natural Science Foundation of China (71231007).

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.